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# Modification of the model code for the ensemble integration

#### Contents

## Overview

Numerical models are typically implemented for normal integration of some initial state. For the data assimilation with filter algorithm, an ensemble of model states has to be integrated for limited time until observations are available and an analysis step of the filter is computed. Subsequently, the updated ensemble has to be integrated again. To allow for these alternating ensemble integrations and analysis steps the model code has to be extended. The recommended implementation strategy for PDAF is to add an additional loop outside of the regular time-stepping loop of the model. This strategy has the potential to reduce the required chances in the model code to the minimum. In addition, a routine that simulates model errors might be required to be inserted into the time stepping loop of the model. The required extensions are described below.

Some operations that are specific to the model and the observations that are assimilated are performed by routines that are supplied by the user and that are called through the defined interface of PDAF. Generally, these user-supplied routines have to provide quite elementary operations, like initializing a model state vector for PDAF from model fields or providing the vector of observations. PDAF provides examples for these routines and templates that can be used as the basis for the implementation. As only the interface of these routines is specified, the user can implement the routines like a routine of the model. Thus, the implementation of these routines should not be difficult.

## External ensemble loop

The external loop for the ensemble integration has to enclose the time stepping loop of the model. Next to the external loop, a control structure for exiting the external loop as well as two calls to subroutines of PDAF have to be added. These are the calls to `PDAF_get_state`

and a filter-specific routine like `PDAF_put_state_seik`

for the SEIK filter. Both routines are described in sections below.

The extended model code can look like this for the SEIK filter:

pdaf_modelloop: DO CALL PDAF_get_state(nsteps, ..., doexit, ...) ! Check whether forecast has to be performed ifcontrol: IF (doexit /= 1) THEN IF (nsteps > 0) THEN ... Time stepping code of the model ... END IF CALL PDAF_put_state_seik(...) ELSE ifcontrol ! No more assimilation work; exit loop EXIT pdaf_modelloop END IF ifcontrol END DO pdaf_modelloop

In this example, which is taken from the example implementation in `testsuite/src/dummymodel_1D`

, we use an unconditional DO loop (while loop). The exit flag `doexit`

for this loop is set within `PDAF_get_state`

. In addition, the variable `nsteps`

is initialized, which defines the number of time steps to be performed during the current forecast phase. Thus, we only execute the time stepping code if `nsteps>0`

. (If this has to be implemented using an IF-clause as in the example should be checked for the particular code).

## PDAF_get_state

The routine `PDAF_get_state`

has the purpose to initialize the information, whether further model integrations have to be computed and how many time steps have to be performed. In addition, the model fields to be propagated are initialized from the array holding the ensemble states.

The interface of `PDAF_get_state`

is the following:

SUBROUTINE PDAF_get_state(nsteps, timenow, doexit, U_next_observation, U_distribute_state, & U_prepoststep, status)

with the following arguments:

`nsteps`

: An integer specifying upon exit the number of time steps to be performed`timenow`

: A real specifying upon exit the current model time.`doexit`

: An integer variable defining whether the assimilation process is completed and the program should exit the while loop. For compatibility 1 should be used for exit, 0 for continuing in the loop.`U_next_observation`

: The name of a user supplied routine that initializes the variables`nsteps`

,`timenow`

, and`doexit`

`U_distributed_state`

: The name of a user supplied routine that initializes the model fields from the array holding the ensemble of model state vectors`U_prepoststep`

: The name of a user supplied routine that is called before and after the analysis step. Here the user has the possibility to access the state ensemble and can e.g. compute estimated variances or can write the ensemble states the state estimate into files.`status`

: The integer status flag. It is zero, if PDAF_get_state is existed without errors.

## PDAF_put_state_X

There is a separate routine `PDAF_put_state_*`

for each of the filter algorithms. The name of the routine includes the name of the filter at its end. The purpose of the `PDAF_put_state_*`

routines is to write back the forecast model fields into the array holding the ensemble of model state vectors. In addition, the routine checks if the current forecast phase is completed. If not, the routine is exited and the next cycle of the ensemble loop is performed. If the current forecast phase is completed, the routine executes the analysis step of the chosen filter algorithm. The interface to each put-state routine is specific for each filter algorithm, because the names of several user-supplied routines have to be specified, which are specific for each filter algorithm. For example, the interface when using the SEIK filter is the following:

SUBROUTINE PDAF_put_state_seik(U_collect_state, U_init_dim_obs, U_obs_op, & U_init_obs, U_prepoststep, U_prodRinvA, U_init_obsvar, status)

with the following arguments:

`U_collect_state`

: The name of the user-supplied routine that initializes a state vector from the array holding the ensembel of model states from the model fields. This is basically the inverse operation to`U_distribute_state`

used in`PDAF_get_state`

`U_init_dim_obs`

: The name of the user-supplied routine that provides the size of observation vector`U_obs_op`

: The name of the user-supplied routine that acts as the observation operator on some state vector`U_init_obs`

: The name of the user-supplied routine that initializes the vector of observations`U_prepoststep`

: The name of the pre/poststep routine as in`PDAF_get_state`

`U_prodRinvA`

: The name of the user-supplied routine that computes the product of the inverse of the observation error covariance matrix with some matrix provided to the routine by PDAF. This operation occurs during the analysis step of the SEIK filter.`U_init_obsvar`

: The name of the user-supplied routine that provides a mean observation error variance to PDAF (This routine will only be executed, if an adaptive forgetting factor is used)`status`

: The integer status flag. It is zero, if PDAF_get_state is exited without errors.

## User-supplied routines

Here, one the user-supplied routines are discussed that are required at this stage of the implementation (that is, the ensemble integration). The routines that are required to conduct the analysis step of some filter, are described in the main section about the implementation of the analysis step. For testing (see Compilation and testing), all routines need to exist, but only those described here in detail need to be implemted with functionality.

To indicate user-supplied routines we use the prefix `U_`

. In the template directory `templates/`

these routines are provided in files with the routines name without this prefix. In the example implementation in `testsuite/src/dummymodel_1D`

the routines exist without the prefix, but with the extension `_dummy_D.F90`

. In the section titles below we provide the name of the template file in parentheses.

`U_next_observation`

(next_observation.F90)

The interface for this routine is

SUBROUTINE U_next_obs(stepnow, nsteps, doexit, timenow) INTEGER, INTENT(in) :: stepnow ! Number of the current time step INTEGER, INTENT(out) :: nsteps ! Number of time steps until next obs INTEGER, INTENT(out) :: doexit ! Whether to exit forecasting (1 for exit) REAL, INTENT(out) :: timenow ! Current model (physical) time

The routine is called once at the beginning of each forecast phase. It is executed by all processes that participate in the model integrations.

Based on the information of the current time step, the routine has to define the number of time steps `nsteps`

for the next forecast phase. In addition, the flag `doexit`

has to be initialized to provide the information if the external ensemble loop can be exited. `timenow`

is the current model time. This variable should also be initialized. It is particularly important, if an ensemble task integrates more than one model state. In this case `timenow`

can be used to correctly jump back in time.

Some hints:

- If the time interval between successive observations is known,
`nsteps`

can be simply initialized by dividing the time interval by the size of the time step `doexit`

should be 0 to continue the assimilation process. In most cases`doexit`

is set to 1, when`PDAF_get_state`

is called after the last analysis for which observations are available.- At the first call to
`U_next_obs`

the variable`timenow`

should be initialized with the current model time. At the next call a forecast phase has been completed. Thus, the new value of`timenow`

follows from the timer interval for the previous forecast phase.

`U_distribute_state`

(distribute_state.F90)

The interface for this routine is

SUBROUTINE distribute_state(dim_p, state_p) INTEGER, INTENT(in) :: dim_p ! State dimension for PE-local model sub-domain REAL, INTENT(inout) :: state_p(dim_p) ! State vector for PE-local model sub-domain

This routine is called during the forecast phase as many times as there are states to be integrated by a model task. Again, the routine is executed by all processes that belong to model tasks.

When the routine is called a state vector `state_p`

and its size `dim_p`

are provided. As the user has defined how the model fields are stored in the state vector, one can initialize the model fields from this information. If the model is not parallelized, `state_p`

will contain a full state vector. If the model is parallelized using domain decomposition, `state_p`

will contain the part of the state vector that corresponds to the model sub-domain for the calling process.

Some hints:

- If the state vector does not include all model fields, it can be useful to keep a separate array to store those additional fields. This array has to be kept separate from PDAF, but can be defined using a module like
`mod_assimilation`

.

`U_prepoststep`

(prepoststep_seik.F90)

The interface of the routine is identical for all filters. However, the particular operations that are performed in the routine can be specific for each filter algorithm. Here, we exemplify the interface on the example of the SEIK filter.

The interface for this routine is

SUBROUTINE prepoststep(step, dim_p, dim_ens, dim_ens_p, dim_obs_p, & state_p, Uinv, ens_p, flag) INTEGER, INTENT(in) :: step ! Current time step ! (When the routine is called before the analysis -step is provided.) INTEGER, INTENT(in) :: dim_p ! PE-local state dimension INTEGER, INTENT(in) :: dim_ens ! Size of state ensemble INTEGER, INTENT(in) :: dim_ens_p ! PE-local size of ensemble INTEGER, INTENT(in) :: dim_obs_p ! PE-local dimension of observation vector REAL, INTENT(inout) :: state_p(dim_p) ! PE-local forecast/analysis state ! The array 'state_p' is not generally not initialized in the case of SEIK. ! It can be used freely here. REAL, INTENT(inout) :: Uinv(dim_ens-1, dim_ens-1) ! Inverse of matrix U REAL, INTENT(inout) :: ens_p(dim_p, dim_ens) ! PE-local state ensemble INTEGER, INTENT(in) :: flag ! PDAF status flag

The routine `U_prepoststep`

is called once at the beginning of the assimilation process. In addition, it is called during the assimilation cycles before the analysis step and after the ensemble transformation. The routine is called by all filter processes (that is `filterpe=1`

).

The routine provides for the user the full access to the ensemble of model states. Thus, user-controlled pre- and post-step operations can be performed. For example the forecast and the analysis states and ensemble covariance matrix can be analyzed, e.g. by computing the estimated variances. In addition, the estimates can be written to disk.

Hint:

- If a user considers to perform adjustments to the estimates (e.g. for balances), this routine is the right place for it.

`U_collect_state`

(collect_state.F90)

The interface for this routine is

SUBROUTINE collect_state(dim_p, state_p) INTEGER, INTENT(in) :: dim_p ! State dimension for PE-local model sub-domain REAL, INTENT(inout) :: state_p(dim_p) ! State vector for PE-local model sub-domain

This routine is called during the forecast phase as many times as there are states to be integrated by a model task. It is called at the end of the integration of a member state of the ensemble. The routine is executed by all processes that belong to model tasks.

When the routine is called, a state vector `state_p`

and its size `dim_p`

are provided. The operation to be performed in this routine is inverse to that of the routine `U_distribute_state`

. That is, the state vector `state_p`

has to be initialized from the model fields. If the model is not parallelized, `state_p`

will contain a full state vector. If the model is parallelized using domain decomposition, `state_p`

will contain the part of the state vector that corresponds to the model sub-domain for the calling process.

Some hints:

- If the state vector does not include all model fields, it can be useful to keep a separate array to store those additional fields. This array has to be kept separate from PDAF, but can be defined using a module like
`mod_assimilation`

.

## Simulating model errors

The implementation of the filter algorithms does not support the specification of a model error covariance matrix. This was left out, because in the SEEK and SEIK filter, the handling can be extremely costly, as the model error covariance matrix has to be projected onto the ensemble space. Instead PDAF support the simulation of model errors by disturbing fields during the model integration. For this, some routine will be required that is inserted into the time stepping loop of the model. As this procedure is specific to each model, the is no routine provided by PDAF for this.

## Compilation and testing

To compile the extended model code with PDAF, one has to extend the Makefile for the model by adding the additional user-supplied routines. While all of the user-supplied routines need to exist not all of them need to be fully implemented at this time if the following procedure is used.

At this implementation stage one can use the preprocessor definition `PDAF_NO_UPDATE`

(available from Version 1.6.1). With this, the actual analysis step of the chosen filter algorithm is not executed. Accordingly, only the user-supplied routines used in `PDAF_get_state`

as well as the routine `U_collect_state`

need to be implemented with functionality. The other routines will not be executed, because they are only called during the analysis step. Generally with `PDAF_NO_UPDATE`

the program performs just an ensemble integration. That is, PDAF is initialized by `PDAF_init`

. Then a forecast is computed by using `PDAF_get_state`

and the chosen `PDAF_put_state_*`

routine. At the initial time `U_prepoststep`

is executed by `PDAF_get_state`

. `U_next_obs`

will provide the number of time steps to be computed by the model and `U_distributed_state`

will initialize the model fields. Subsequently the ensemble integration is performed and the forecast fields are written back to the ensemble array by `U_collect_state`

. Upon completion of the forecast phase, the routine `U_prepoststep`

is executed twice. The first time is the regular call before the analysis is executed. Thus, it allows to access the forecast ensemble. If the analysis would not be deactivated, the second call to `U_prepoststep`

would be after the analysis allowing access to the ensemble directly after the analysis. As the analysis is deactivated here, the ensemble will be the same as in the first call.

This test allows to check the following:

- Is
`U_prepoststep`

working correctly? - Does
`U_next_obs`

work correctly and is the information from this routine used correctly for the model integration - Are
`U_distribute_state`

and`U_collect_state`

work correctly?

One could also comment out the actual time stepping part of the model. This would allow to only test the interfacing between PDAF and the model.

It is important to ensure that the ensemble integration performs correctly. The simplest case should be a parallel configuration in which the number of model tasks equals the ensemble size as here the model tasks always compute forward in time. If the number of model tasks is smaller than the ensemble size, some model tasks will have to integrate multiple states of the ensemble. If a model task has to integrate two states, the model will have to jump back in time for the integration of the second state. It might be that some arrays of the model need to be re-initialized to ensure that the second integration is consistent. Also, one might need to check if the initialization of forcing fields (e.g. wind stress over the ocean) performs correctly for the second integration. (Sometimes model are implemented with the constraint that the model time always increases, which is the normal case for pure model simulations without assimilation.) A useful test is to initialize an ensemble in which all states are equal. If this ensemble is integrated the forecast states of the ensemble should, of course, still be equal.